生产率与效率分析lecture8-随机前沿效率1

TheStochasticFrontierAnalysis(SFA)Slide#2TheStochasticFrontier-HistoryFarrell(1957)2measuresoffirmefficiencytechnicalefficiency(TE)obtainmaximaloutputfromgivensetofinputsallocativeefficiency(AE)useinputsinoptimalproportions,giveninputpricesSlide#3History(cont.)Recalltechnicalefficiency(TE)allocativeefficiency(AE)FocusonTEonlyForsimplicityinexplanationofstochasticfrontiers(SF)Slide#4History(cont.)TocalculateefficiencymustknowproductionfunctionoffullyefficientfirmNeverknowitFarrellestimatefullyefficientproductionfunctionSlide#5History(cont.)EstimatefullyefficientproductionfunctionTwoways1.non-parametricpiece-wiselineartechnologyCharnes,CooperothersDEA(notDrugEnforcementAgency!)ignoreinthissessionSlide#6History(cont.)EstimatefullyefficientproductionfunctionTwoways1.non-parametricpiece-wiselineartechnologyCharnes,CooperothersDEA(DataEnvelopmentAnalysis)ignoreinthissessionSlide#7History(cont.)EstimatefullyefficientproductionfunctionTwoways2.parametricfunctionAigner,LovellandSchmidt,othersstochasticfrontiermodelSlide#8History(cont.)Consideraproductionfunctionlnyi=xii=1,2,….,Nlnyiislogofscalaroutputfori-thfirmxiisrowvectoroflogsofKinputsbyi-thfirmiscolumnvectorofKunknowncoefficientsFirmsattempttomaximizetheobservedoutput(lny)producedbyinputs(x)giventechnologyandcircumstancesSlide#9History(cont.)lnyi=xii=1,2,….,Nso,observedoutputshould=xicallxi“frontieroutput”=yFusuallyobservedoutput(yO)xibecauseofinefficiency(&otherreasons)Slide#10History(cont.)So,usualcaseisobservedoutputfrontieroutputyOyFlnyixiThatis,firmisinefficientSlide#11History(cont.)RecalllnyixiAigner&Chu(1968)lnyi=xi-uii=1,2,….,Nuinon-negativerandomvariablecapturestechnicalinefficiency(TI)inproductionini-thfirmSlide#12Aigner&Chu(cont.)Fori-thfirm,givenxiTEiisobservedoutput/frontieroutputyi/exp(xi)=exp(xi-ui)/exp(xi)=exp(-ui)Here,0exp(-ui)1Slide#13Aigner&Chu(cont.)TEiisobservedoutput/frontieroutput=exp(-ui)；0exp(-ui)1magnitudeofi-thfirm’sobservedoutputrelativetowhatcouldbeproducedbyfullyefficientfirmusingsamexvectorSlide#14History(cont.)Whatproblemdoyouseewiththismodel?(What’smissing?)lnyi=xi-uii=1,2,….,Nuinon-negativerandomvariablecapturestechnicalinefficiency(TI)inproductionini-thfirmSlide#15Problem(cont.)lnyi=xi-uiuassumedtomeasureonlyextentofinefficiencyBUT...ucanalsocapturemeasurementerroriny&othernoiseSo,addanotherdisturbancetermlnyi=xi+vi-uivi.i.d.N(0,v2)u0Slide#16StochasticFrontierlnyi=xi+vi-uivi.i.d.N(0,v2)u0v&uindependentvaccountsformeasurementerrorrandomfactorssuchasweatherstrikesluck...Slide#17StochasticFrontier(cont.)lnyi=xi+vi-uivi.i.d.N(0,v2)u0v&uindependentumeasuresTEuassumedi.i.d.exponentialorhalf-normalSlide#18StochasticFrontier(cont.)yF(frontieroutput):xi+viyO(observedoutput):lnyi=xi+vi-uiNOTE:called“stochastic”frontiermodelbecauseyFvariesoverfirmsyOboundedabovebystochastic(random)yFSlide#19StochasticFrontier(cont.)Basicfeaturesofstochasticfrontier(SF)illustratedinFig.1inputsonhorizontalaxisoutputsonverticalaxisassumingdiminishingreturnstoscaleapplyobservedinputsandoutputsfortwofirms,i&jSlide#20yxxixJyiyJdeterministicproductionfunctiony=exp(x)Slide#21yxxixJyiyJdeterministicproductionfunctiony=exp(x)frontieroutputiexp(xi+vi),ifvi0observedoutputiexp(xi+vi-ui)Slide#22yxxixJyiyJdeterministicproductionfunctiony=exp(x)frontieroutputjexp(xJ+vJ),ifvJ0frontieroutputiexp(xi+vi),ifvi0observedoutputjexp(xJ+vJ-uJ)observedoutputiexp(xi+vi-ui)Slide#23StochasticFrontier(cont.)Composederror(vi-ui)causesnoproblemsinestimationOLSbunbiased,consistent,efficientamonglinearestimatorsexceptestimatorofinterceptnotconsistentMLestimatoryieldsmoreefficientb(thanOLS)consistentinterceptconsistentvarianceof(vi-ui)Slide#24StochasticFrontier(cont.)Aigner,Lovell,Schmidt(1977)derivedlog-likelihoodfunctionformodelwithcomposederror(vi-ui)vi.i.d.N(0,v2)ui.i.d.truncations(atzero)N(0,u2)“half-normal”varianceparameters2=v2+u2=[u/v]0Slide#25StochasticFrontier(cont.)BatteseandCora(1977)reparameterizedlog-likelihoodfunctionwithcomposederror(vi-ui)varianceparameters2=v2+u2=(u2/2)[0,1]vs.=[u/v]0Whatiscausingdeviationsfromfrontierwhen=0?DeviationsfromfrontierdueentirelytonoiseSlide#26StochasticFrontier(cont.)BatteseandCora(1977)reparameterizedlog-likelihoodfunctionwithcomposederror(vi-ui)varianceparameters2=v2+u2=(u2/2)[0,1]vs.=[u/v]0Whatiscausingdeviationsfromfrontierwhen=1?DeviationsfromfrontierdueentirelytoinefficiencySlide#27BatteseandCora(cont.)varianceparameters2=v2+u2=(u2/2)[0,1]thisparameterizationhasadvantagethatcansearchforvaluesofover[0,1]asstartvalueforiterativemaximizationstepSlide#28BatteseandCora(cont.)lnL=-(N/2)ln(/2)-(N/2)ln(2)+ln[1-(zi)]-(1/22)(lnyi-xi)2wherezi=[(lnyi-xi)/](/1-)and(.)isdistributionfunctionofN(0,1)maximizelnLtoobtainMLestimatesof,2and(K+2)Slide#29BatteseandCora(cont.)Parameters:,2andThreestepsofmaximization1.OL